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##  Improving Student Achievement in Mathematics, Part 1: Research                  Findings. ERIC Digest.                                                       

The number of research studies conducted in mathematics education               over the past three decades has increased dramatically (Kilpatrick,             1992). Research findings indicate that certain teaching strategies              and methods are worth careful consideration as teachers strive to               improve their mathematics teaching practices.  For the classroom                implications of the research findings summarized here, please see               the companion to this Digest,  "Improving Student Achievement in                Mathematics, Part 2: Recommendations for the                                    Classroom" (EDO-SE-00-10) 
1. The extent of the students'                      opportunity to learn mathematics content bears directly and                     decisively on student mathematics achievement. 
Opportunity to                 learn (OTL) was studied in the First International Mathematics                  Study (Husen, 1967), where teachers were asked to rate the extent of            student exposure to particular mathematical concepts and skills.                Strong correlations were found between OTL scores and mean student              achievement scores, with high OTL scores associated with high                   achievement.  The link was also found in subsequent international               studies, such as the Second International Mathematics                           Study (McKnight et al., 1987) and the Third International                       Mathematics and Science study (TIMSS) (Schmidt, McKnight, & Raizen,             1997). 
2.  Focusing instruction on the meaningful development of              important mathematical ideas increases the level of student                     learning. 
There is a long history of research, going back to the              work of Brownell (1945,1947), on the effects of teaching for meaning            and understanding. Investigations have consistently shown that an               emphasis on teaching for meaning has positive effects on student                learning, including better initial learning, greater retention and              an increased likelihood that the ideas will be used in new                      situations. 
3.  Students can learn both concepts and skills by                solving problems. 
Research suggests that students who develop                 conceptual understanding early perform best on procedural knowledge             later.  Students with good conceptual understanding are able to                 perform successfully on near-transfer tasks and to develop                      procedures and skills they have not been taught. Students with low              levels of conceptual understanding need more practice in order to               acquire procedural knowledge. 
4.  Giving students both an                     opportunity to discover and invent new knowledge and an opportunity             to practice what they have learned improves student                             achievement. 
Data from the TIMSS video study show that over 90% of            mathematics class time in the United States 8th-grade classrooms is             spent practicing routine procedures, with the remaining time                    generally spent applying procedures in new situations.  Virtually no            time is spent inventing new procedures and analyzing unfamiliar                 situations.  In contrast, students at the same grade level in                   typical Japanese classrooms spend approximately 40% of instructional            time practicing routine procedures, 15% applying procedures in new              situations, and 45% inventing new procedures and analyzing new                  situations. 
Research suggests that students need opportunities for            both practice and invention.  Findings from a number of studies show            that when students discover mathematical ideas and invent                       mathematical procedures, they have a stronger conceptual                        understanding of connections between mathematical ideas. 
5.                   Teaching that incorporates students' intuitive solution methods can             increase student learning, especially when combined with                        opportunities for student interaction and discussion. 
Student                 achievement and understanding are significantly improved when                   teachers are aware of how students construct knowledge, are familiar            with the intuitive solution methods that students use when they                 solve problems, and utilize this knowledge when planning and                    conducting instruction in mathematics. 
Structuring instruction                around carefully chosen problems, allowing students to interact when            solving problems, and then providing opportunities for them to share            their solution methods result in increased achievement on                       problem-solving measures.  These gains come without a loss of                   achievement in the skills and concepts measured on standardized                 achievement tests. 
6.  Using small groups of students to work on              activities, problems and assignments can increase student                       mathematics achievement. 
Davidson (1985) reviewed studies that                compared student achievement in small group settings with                       traditional whole-class instruction.  In more than 40% of these                 studies, students in the classes using small group approaches                   significantly outscored control students on measures of student                 performance. In only two of the 79 studies did control-group                    students perform better than the small group students, and in these             studies there were some design irregularities. From a review of 99              studies of cooperative group-learning methods, Slavin (1990)                    concluded that cooperative methods were effective in improving                  student achievement.  The most effective methods emphasized both                group goals and individual accountability. 
7. Whole-class                     discussion following individual and group work improves student                 achievement. 
Research suggests that whole class discussion can be             effective when it is used for sharing and explaining the variety of             solutions by which individual students have solved problems.  It                allows students to see the many ways of examining a situation and               the variety of appropriate and acceptable solutions. Wood (1999)                found that whole-class discussion works best when discussion                    expectations are clearly understood.  Students should be expected to            evaluate each other's ideas and reasoning in ways that are not                  critical of the sharer.  Students should be expected to be active               listeners who participate in discussion and feel a sense of                     responsibility for each other's understanding. 
8. Teaching                    mathematics with a focus on number sense encourages students to                 become problem solvers in a wide variety of situations and to view              mathematics as a discipline in which thinking is                                important. 
"Number sense" relates to having an intuitive feel for             number size and combinations, and the ability to work flexibly with             numbers in problem situations in order to make sound decisions and              reasonable judgments.  It involves mentally computing, estimating,              sensing number magnitudes, moving between representation systems for            numbers, and judging the reasonableness of numerical results.                   Markovits and Sowder (1994) studied 7th-grade classes where special             units on number magnitude, mental computation and computational                 estimation were taught.  They determined that after this special                instruction, students were more likely to use strategies that                   reflected sound number sense, and that this was a long-lasting                  change. In a study of second graders, Cobb (1991) and his colleagues            found that students' number sense was improved by a problem-centered            curriculum that emphasized student interaction and self-generated               solution methods.  Almost every student developed a variety of                  strategies to solve a wide range of problems.  Students also                    demonstrated increased persistence in solving problems. 
9.                    Long-term use of concrete materials is positively related to                    increases in student mathematics achievement and improved attitudes             towards mathematics. 
In a review of activity-based learning in                mathematics in kindergarten through grade 8, Suydam and                         Higgins (1977) concluded that using manipulative materials produces             greater achievement gains than not using them.  In a more recent                meta-analysis of sixty studies (kindergarten through postsecondary)             that compared the effects of using concrete materials with the                  effects of more abstract instruction, Sowell (1989) found that the              long-term use of concrete materials by teachers knowledgeable in                their use improved student achievement and attitudes. 
10. Using               calculators in the learning of mathematics can result in increased              achievement and improved student attitudes. 
Studies have                      consistently shown that thoughtful use of calculators improves                  student mathematics achievement and attitudes toward mathematics.               From a meta-analysis of 79 non-graphing calculator studies, Hembree             and Dessart (1986) concluded that use of hand-held calculators                  improved student learning.  Analysis also showed that students using            calculators tended to have better attitudes towards mathematics and             better self-concepts in mathematics than their counterparts who did             not use calculators.  They also found that there was no loss in                 student ability to perform paper-and-pencil computational skills                when calculators were used as part of mathematics                               instruction. 
Research on the use of graphing calculators has also             shown positive effects on student achievement.  Most studies have               found positive effects on students' graphing ability, conceptual                understanding of graphs and their ability to relate graphical                   representations to other representations.  Most studies of graphing             calculators have found no negative effect on basic skills, factual              knowledge, or computational skills. REFERENCESBrownell,                   W.A. (1945). When is arithmetic meaningful?  "Journal of Education              Research," 38, 481-98. 
Brownell, W.A.  (1947). The place of                   meaning in the teaching of arithmetic.  "Elementary School Journal,"            47, 256-65. 
Cobb, P, et al.  (1991). Assessment of a                          problem-centered second-grade mathematics project.  "Journal for                Research in Mathematics Education," 22, 3-29. 
Davidson, N. (1985).            Small group cooperative learning in mathematics: A selective view of            the research.  In R. Slavin (Ed.), "Learning to cooperate:                      Cooperating to learn." (pp.211-30) NY: Plenum. 
Hembree, R. &                  Dessart, D.J. ((1986). Effects of hand-held calculators in                      pre-college mathematics education: A meta-analysis.  "Journal for               Research in Mathematics Education," 17, 83-99. 
Husen,                         T. (1967). "International study of achievement in mathematics," Vol.            2. NY: Wiley. 
Kilpatrick, J.  (1992).  A history of research in               mathematics education. In Grouws, D. A., (Ed.), "Handbook of                    research on mathematics teaching and learning." (pp. 3-38) NY:                  Macmillan. 
Markovit, Z., & Sowder, J.  (1994). Developing number              sense: An intervention study in grade 7.  "Journal for Research in              Mathematics Education," 25, 4-29. 
McKnight, I.V.S., et                        al. (1987). "The underachieving curriculum." Champaign, IL:                     Stipes. 
Schmidt, W.H., McKnight, C.C., & Raizen, S.A. (1997). "A              splintered vision: An investigation of U.S. science and mathematics             education." Dordrecht, Netherlands: Kluwer. 
Slavin, R.E.  (1990).             Student team learning in mathematics. In N.                                     Davidson (Ed.), "Cooperative learning in math: A handbook for                   teachers". Boston: Allyn & Bacon, (pp. 69-102). 
Sowder,                       J. (1992a). Estimation and number sense.  In D.A.                               Grouws (Ed.), "Handbook of research on mathematics teaching and                 learning." (pp. 371-89) NY: Macmillan. 
Sowder, J. (1992b). Making             sense of numbers in school mathematics. In R. Leinhardt, R.                     Putman, & R. Hattrup (Eds.), "Analysis of arithmetic for mathematics            education." (pp. 1-51)Hillsdale, NJ: Lawrence Erlbaum. 
Sowell,                E.J. (1989). Effects of manipulative materials in mathematics                   instruction. "Journal for Research in Mathematics Education," 20,               498-505. 
Suydam, M.N. & Higgins, J. L.  (1977).  "Activity-based              learning in elementary school mathematics: Recommendations from                 research."  Columbus, OH: ERIC Clearinghouse for Science,                       Mathematics, and Environmental Education. 
Wood, T.  (1999).                   Creating a context for argument in mathematics class. "Journal for              Research in Mathematics Education," 30, 171-91. 
Other Sources of              Information About Best Practices: 
NCTM                                        Illuminations (http://illuminations.nctm.org/index2.html) 
National            Center for Improving Student Learning and Achievement in Mathematics            and Science (http://www.wcer.wisc.edu/ncisla/) 
Eisenhower National            Clearinghouse for Mathematics and Science                                       Education (http://enc.org) 
An expanded version of the ideas                   presented in this Digest is available online at                                 http://www.ibe.unesco.org/Publications/Practice/prac04e.pdf 
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This digest was funded by the Office of Educational Research and              Improvement, U.S. Department of Education, under contract no.                   ED-99-CO-0024. Opinions expressed in this digest do not necessarily             reflect the positions or policies of OERI or the U.S. Department of             Education.                                                                   **Title:** Improving Student Achievement in Mathematics, Part 1: Research                  Findings. ERIC Digest.                                                       **Note:** For Part 2, see SE 064 318.                                                  **Document Type:** Information Analyses---ERIC Information Analysis Products (IAPs) (071); Information Analyses---ERIC Digests (Selected) in Full Text (073); **Available From:** ERIC/CSMEE, 1929 Kenny Road, Columbus, OH 43210-1080. Tel:                      800-276-0462 (Toll Free); e-mail: [email protected]; Web site:                     http://www.ericse.org.                                                       **Descriptors:**  Academic Achievement, Calculators, Concept Formation, Elementary               Secondary Education,  Mathematics Education, Problem Solving                 **Identifiers:** ERIC Digests                                                                 
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